Logic Seminar: 09/1/2017 - 03/20/2019

Abstract. In their 1985 paper, Abraham, Rubin, and Shelah studied a number of combinatorial principles about \(\aleph_1\)-sized objects. One such axiom is the so-called "ARS Open Coloring Axiom" (hereafter \(\mathsf{OCA}_{ARS}\)), which concerns decompositions of \(\aleph_1\)-sized sets of reals into homogeneous sets for clopen colorings. One of the main open questions from their paper is whether or not \(\mathsf{OCA}_{ARS}\) is consistent with a value of the continuum greater than \(\aleph_2\) (it implies that the continuum is at least \(\aleph_2\)).

There are two additional theorems which complicate the situation. First, Moore has shown that Todorcevic's Open Coloring Axiom (hereafter \(\mathsf{OCA}_T\)) together with \(\mathsf{OCA}_{ARS}\) decides the value of the continuum to be exactly \(\aleph_2\), but second, Farah has shown that (a restricted version of) \(\mathsf{OCA}_T\) is consistent with an arbitrarily large value of the continuum. It is therefore of interest whether or not \(\mathsf{OCA}_{ARS}\) on its own decides the value of \(2^{\aleph_0}\).

Recently Gilton and Itay Neeman have answered this question, showing that \(\mathsf{OCA}_{ARS}\) is in fact consistent with \(2^{\aleph_0}=\aleph_3\). As in the original ARS paper, we need to build so-called preassignments of colors in order to add the requisite homogeneous sets. However, these can only be built over models satisfying the \(\mathsf{CH}\). To get around this difficulty, we build preassignments with very strong symmetry conditions, which allow us to combine them in many different ways, using a new type of poset called a partition product. In this talk, we will motivate and define partition products, sketch the construction of these preassignments, and show how, as a result, we can obtain a model of \(\mathsf{OCA}_{ARS}\) with a large value of the continuum.

Abstract. A measure-theoretic sheaf is constructed over an arbitrary sigma-finite measure space. The collection of such sheaves forms a topos, and is an instance of a Boolean-valued model of ZFC. In this talk, adopting a variant of the sheaf-theoretic approach, we present semantics of some fundamental structures in this model such as vector spaces, topological spaces, and measure spaces. Interestingly, it turns out that these Boolean-valued structures allow for a meaningful interpretation in a standard model as well which often can be characterized by kernel type objects. This opens a way for applications of Boolean-valued techniques to analysis and its applications. We present details on two of these applications in probability and ergodic theory.

Abstract. The connection between finitely additive probability measures and NIP theories was first noticed by Keisler. Around 20 years later, the work of Hrushovski, Peterzil, Pillay, and Simon greatly expanded this connection. Out of this research came the concept of generically stable measures. In the context of NIP theories, these particular measures exhibit stable behavior. In particular, Hrushovski, Pillay, and Simon demonstrated that generically stable measures admit a natural finite approximation. In this talk, we will discuss generically stable measures in the local setting. We will describe connections between these measures and concepts in functional analysis as well as show that this interpretation allows us to derive an approximation theorem.

Abstract. An active line of research in modern combinatorics is extending classical results from the dense setting (e.g., Szemeredi's theorem) to the sparse random setting. These results state that a property of a given "dense" structure is inherited by a randomly chosen "sparse" substructure. A recent breakthrough tool for proving such statements is the Balogh-Morris-Samotij and Saxton-Thomason hypergraph containers method, which bounds the number of independent sets in finite hypergraphs. In a joint work with A. Bernshteyn, M. Delcourt, and H. Towsner, we give a new — elementary and nonalgorithmic — proof of the containers theorem for finite hypergraphs. Our proof is inspired by considering hyperfinite hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. Applying this intuition in another setting with a notion of dimension, namely, algebraically closed fields, A. Bernshteyn, M. Delcourt, and I prove an analogous theorem for "dense" algebraically definable hypergraphs: any Zariski-generic low-dimensional subset of such hypergraphs is itself "dense" (in particular, not independent).

Abstract. We show the consistency of $\square_{\kappa, 2}$ plus $SATP(\kappa^+)$ for regular $\kappa$ assuming a weakly compact. Using methods of Golshani-Hayut we also establish a global consistency result for successors of regulars from class many supercompacts.

Abstract. Let T be an o-minimal theory extending that of ordered abelian groups. In joint work with Hieronymi, we showed that the theory of dense pairs of models of T is non-distal. I will now discuss the connection between distal and non-distal types in this theory to the notion of Large Dimension, which arises from a suitable pre-geometry.

Abstract. In this talk I will first define and describe the mathematical object $\mathbb{T}_{\log}$: the ordered valued differential field of logarithmic transseries. I will then discuss a strategy I have developed for proving $\mathbb{T}_{\log}$ is model complete in a certain language that I will introduce. I reduce the problem of model completeness down to a few precise conjectures about the nature of logarithmic derivatives, logarithms, solutions of linear differential equations, and differential-transcendence. Recent progress made in the last year will also be mentioned.

Abstract. We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models of set theory without choice, and apply it to solve questions of Hjorth-Kechris-Louveau.
E.g., we show that $\Sigma^0_{\omega+1}$ equivalence relations induced by abelian group actions are strictly simpler than general $\Sigma^0_{\omega+1}$ orbit equivalence relations.
The proof goes through studying symmetric models generated by generic invariants of these equivalence relations.
We will use models which were constructed by G. Monro in 1973 to separate the ``generalized Kinna-Wagner principles'', $\mathrm{KWP}^n$.
We show that these models correspond to the irreducibilities along the finite Friedman-Stanley jumps, $=^{+n}<_B =^{+(n+1)}$.
We extend Monro's results through limit stages, thus showing the consistency of $\mathrm{KWP}^{\omega+1}\wedge\neg\mathrm{KWP}^\omega$, answering a question of Karagila.

Abstract. The Weak Pinsker entropy is an easy-to-define measure-conjugacy invariant of measure-preserving actions of countable groups inspired by the classical Weak Pinsker conjecture, recently proven by Tim Austin for amenable groups. I'll discuss two recent results: (1) (joint with Robin Tucker-Drob) if the group is Bernoulli cocycle-superrigid, then WP-entropy is an orbit-equivalence invariant, (2) WP entropy is bounded by sofic entropy and can be strictly less.

Abstract. It is a major goal of modern set theory to understand the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler's proof does not resemble Kunen's, however, using almost-disjoint coding instead of Kunen's innovative method of coding along branchless trees. We show how to reconcile this gap, giving a new proof of Schindler's theorem that generalizes Kunen's methods and suggests further investigation of non-thin trees. This work is joint with Itay Neeman.

Abstract. The word "generic" is often applied to a theory $T^*$ when it arises as a model companion of a base theory $T$, augmented with extra structure (e.g. a generic automorphism, a generic predicate, a generic order, etc.). Generic theories exhibit lots of "random" behavior, so they are rarely stable or NIP, but they can sometimes be shown to be simple by characterizing a well-behaved notion of independence in $T^*$ (namely non-forking independence) in terms of independence in $T$. Recently, there has been increased interest in the property NSOP1, a generalization of simplicity, spurred by the work of Chernikov, Kaplan, and Ramsey, who showed that NSOP1 theories can also be characterized by the existence of a well-behaved notion of independence (namely Kim independence). In this talk, I will present a number of preservation results for simplicity and NSOP1 under generic constructions, and characterizations of notions of independence in the resulting theories. In joint work with Nicholas Ramsey, generic expansion and generic Skolemization: add new symbols to the language, interpreted arbitrarily or as Skolem functions, and take the model companion. And in very recent results towards a joint project with Minh Chieu Tran and Erik Walsberg, interpolative fusion: given an $L_1$-theory $T_1$ and and $L_2$-theory $T_2$, which intersect in an $L_0$-theory $T_0$, take the model companion of the union of $T_1$ and $T_2$.

Abstract. It has been shown by Goldman (1971) that the open 2-manifolds can be completely classified by algebraic structures. It is known that the open 3-manifolds are more complicated than the open 2-manifolds as witnessed e.g. by the Whitehead manifolds. This presents us with a question: do open 3-manifolds admit classification by countable structures? The conjecture is "no" and the attempt is to use the theory of turbulence developed by Hjorth, Kechris and others to attack this problem. In this talk I will present results surrounding this conjecture and put them in a broader context of descriptive set theory and Borel-reducibility. Finally I will present some new ideas on how to tackle the conjecture of non-classifiability of open 3-manifolds.

Abstract. In 1932 von Neumann proposed classifying the statistical behavior of diffeomorphisms of compact manifolds. Notable progress was made by Halmos and von Neumann using spectral invariants. Later, Bernoulli shifts were classified by Ornstein using Kolmogorov's notion of entropy.

In these talks we show that the general von Neumann problem is intractable in a rigorous sense: the collection of pairs of diffeomorphisms of the torus that are isomorphic is complete analytic — in particular not Borel. It follows that there is no inherently countable procedure for determining whether a given pair (S,T) is isomorphic.

A problem closely related to the isomorphism problem is the Realization Problem: which abstract measure preserving transformations can be realized as diffeomorphisms of a compact manifold. The only known restriction is that the transformations must have finite entropy. As a byproduct of the anti-classification result stated above, we show that a large class of previously unknown examples can be realized smoothly. For example there are measure distal diffeomorphisms of the 2-torus of arbitrary countable ordinal height, and diffeomorphisms of the torus whose simplex of invariant measures is affinely homeomorphic to an arbitrary compact Choquet simplex.

This latter results are proved by establishing a Global Structure Theorem that says that there is a categorical isomorphism between the collection of ergodic transformations with odometer factors and diffeomorphisms realized by the Anosov-Katok method off conjugacy.

Abstract. We will represent the universal Menger curve as a canonical quotient of a projective Fraisse limit
and we will illustrate how various homogeneity and universality properties of this space reduce to standard Fraisse theory and basic combinatorics. Finally we will discuss how our approach can be extended to higher dimensions, where it suggests the existence of homology versions of the $n$-dimensional Menger space, as well as a homology version of the Hilbert cube.

Abstract. The Aldous-Hoover Theorem gives a characterization of those random processes which generate "exchangeable" first-order structures. Exchangeable structures are precisely those where the "labels" of the points do not matter — they are random structures whose distribution remains the same when we permute the points. They therefore correspond to the structures we get when we sample countable substructures randomly from an ultraproduct; indeed, the original proof of the full Aldous-Hoover Theorem used ultraproducts, and the topic remains intimately tied to the way probability measures behave in ultraproducts.

For some purposes, full exchangeability is too strong: we should consider only those permutations respecting some existing structures. A full Aldous-Hoover theorem is not always possible in this setting, and how much we recover turns out to depend on the amalgamation properties of M.

Our goal in this talk is to explain the connection between exchangeability and model theoretic notions, without assuming much prior expertise in either.

Abstract. I will talk about Mutually Stationary sequences of sets.
The notion of Mutual Stationarity was introduced by Foreman and Magidor in the 90s, and has been developed as a notion of stationarity for subsets of singular cardinals and as means to address some classical problems in set theory.
We will discuss the basic concepts and describe several known and recent results.

Abstract. In 1932 von Neumann proposed classifying the statistical behavior of diffeomorphisms of compact manifolds. Notable progress was made by Halmos and von Neumann using spectral invariants. Later, Bernoulli shifts were classified by Ornstein using Kolmogorov's notion of entropy.

In these talks we show that the general von Neumann problem is intractable in a rigorous sense: the collection of pairs of diffeomorphisms of the torus that are isomorphic is complete analytic — in particular not Borel. It follows that there is no inherently countable procedure for determining whether a given pair (S,T) is isomorphic.

A problem closely related to the isomorphism problem is the Realization Problem: which abstract measure preserving transformations can be realized as diffeomorphisms of a compact manifold. The only known restriction is that the transformations must have finite entropy. As a byproduct of the anti-classification result stated above, we show that a large class of previously unknown examples can be realized smoothly. For example there are measure distal diffeomorphisms of the 2-torus of arbitrary countable ordinal height, and diffeomorphisms of the torus whose simplex of invariant measures is affinely homeomorphic to an arbitrary compact Choquet simplex.

This latter results are proved by establishing a Global Structure Theorem that says that there is a categorical isomorphism between the collection of ergodic transformations with odometer factors and diffeomorphisms realized by the Anosov-Katok method off conjugacy.

Abstract. In the early seventies, several decisive relations were noticed between the homological dimension and the cardinality of a small category - particularly for the cardinalities $\aleph_n$ $(n\in\mathbb{N})$. Those relations, in the case of $n=1$, are best understood in terms of Todorcevic's method of minimal walks on the countable ordinals; the higher-$n$-cases point, similarly, to generalizations of that method, and to $n$-dimensional incompactness principles correlated, for each $n$, to $\aleph_n$. All these are ZFC phenomena. How these principles behave on other cardinals is a largely open question. We discuss these matters, and some of their implications both in set theory and in algebraic topology.

Abstract. The notion of n-dependence was introduced by Shelah motivated to treat vector spaces with a bilinear form along NIP theories. One of the important tools analyzing the property is the Ramsey property of the class of ordered finite hyper graphs. Actually many of known results, preservation theorem under boolean combinations, finding a witness in a single variable and characterizing the property by generalized indiscernible can be implied from the Ramsey property. In this point of view we introduce the notion of 2-Order Property, which lies between Independent Property and 2-Independent Property, related a Ramsey class consisting of ladder-like graphs, and demonstrate that how the Ramsey property works well to prove similar results. The typical examples of 2-dependence are not 2-OP. So far, It is open if 2-OP is strictly weaker than 2-IP.

Abstract. (Joint work with Julia Ruppert) Understanding integration in the o-minimal setting is an important and difficult task. By the work of Comte, Lion and Rolin, succeeded by the work of Cluckers and Miller, parameterized integrals of globally subanalytic functions are very well analyzed. But very little is known when the exponential function comes into the game. We consider certain parameterized exponential integrals which come from considering the Brownian motion on globally subanalytic sets. We are able to show nice asymptotic expansions of these integrals.

Abstract. A topological group admits a free action if there is a
compact space on which it acts without fixed points. We translate this property into colorability of graphs, which leads us to questions of combinatorial nature. This is a joint work in progress with Vladimir Pestov.